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(* Title: HOL/UNITY/AllocImpl
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 2000 University of Cambridge
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Implementation of a multiple-client allocator from a single-client allocator
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add_path "../Induct";
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time_use_thy "AllocImpl";
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*)
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AddIs [impOfSubs subset_preserves_o];
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Addsimps [funPair_o_distrib];
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Addsimps [Always_INT_distrib];
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Delsimps [o_apply];
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(*Eliminate the "o" operator*)
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val o_simp = simplify (simpset() addsimps [o_def]);
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Goalw [merge_spec_def,merge_eqOut_def]
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"[| M: merge_spec; G: preserves merge.Out; G: preserves merge.iOut |] \
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\ ==> M Join G : Always {s. length (merge.Out s) = length (merge.iOut s)}";
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by (force_tac (claset() addDs [guaranteesD], simpset()) 1);
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qed "Merge_Always_Out_eq_iOut";
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(*******ALLOCBASE.ML***)
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Goal "[| finite A; finite B; A Int B = {} |] \
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\ ==> bag_of (sublist l (A Un B)) = \
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\ bag_of (sublist l A) + bag_of (sublist l B)";
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by (asm_simp_tac (simpset() addsimps [bag_of_sublist_Un_Int RS sym]) 1);
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qed "bag_of_sublist_Un_disjoint";
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Goal "[| finite I; ALL i:I. finite (A i); \
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\ ALL i:I. ALL j:I. A i Int A j = {} |] \
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\ ==> bag_of (sublist l (UNION I A)) = \
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\ setsum (%i. bag_of (sublist l (A i))) I";
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by (asm_simp_tac (simpset() addcongs [setsum_cong]
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addsimps [bag_of_sublist, setsum_UN_disjoint]) 1);
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qed_spec_mp "bag_of_sublist_UN_disjoint";
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Goal "M : merge_spec \
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\ ==> M : (INT i: lessThan Nclients. Increasing (sub i o merge.In)) \
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\ guarantees[funPair merge.Out merge.iOut] \
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\ (bag_of o merge.Out) Fols \
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\ (%s. setsum (%i. (bag_of o sub i o merge.In) s) \
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\ (lessThan Nclients))";
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by (rtac guaranteesI 1);
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by (res_inst_tac
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[("f", "%s. setsum (%i. bag_of (sublist (merge.Out s)
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{k. k < size(merge.iOut s) &
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nth(merge.iOut s)k = i}))
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(lessThan Nclients)")]
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Always_Follows 1);
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by (rtac UNIV_AlwaysI 1);
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by (Clarify_tac 1);
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by (stac (bag_of_sublist_UN_disjoint RS sym) 1);
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by (force_tac (claset(), simpset() addsimps []) 1);
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by (blast_tac (claset() addIs [bounded_nat_set_is_finite]) 1);
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by (Clarify_tac 1);
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by (simp_tac (simpset() addsimps [o_def]) 1);
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by (asm_full_simp_tac (simpset() delsimps [sublist_upt_eq_take]
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addsimps [bag_of_sublist]) 1);
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by (rtac Follows_trans 1);
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by (rtac Follows_setsum 2);
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by (asm_full_simp_tac (simpset() addsimps []) 3);
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by (Clarify_tac 2);
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by (asm_full_simp_tac (simpset() addsimps [o_assoc RS sym]) 2);
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by (rtac (impOfSubs (mono_bag_of RS mono_Follows_o)) 2);
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(*Now need to invoke the merge_follows guarantee*)
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by (asm_full_simp_tac (simpset() addsimps [merge_spec_def,merge_follows_def]) 2);
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by (res_inst_tac [("f", "%s. bag_of (sublist (merge.Out s) {..length(merge.iOut s)(})")]
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Always_Follows 1);
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by (rtac guaranteesI 1);
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by (res_inst_tac
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[("f", "%s. bag_of (sublist (merge.Out s) {..length(merge.iOut s)(})")]
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Always_Follows 1);
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by (etac (Merge_Always_Out_eq_iOut RS Always_Compl_Un_eq RS iffD1) 1);
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by (force_tac (claset() addIs [UNIV_AlwaysI], simpset() addsimps [o_apply]) 3);
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by (Asm_full_simp_tac 1);
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by (Asm_full_simp_tac 1);
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by (asm_full_simp_tac (simpset() delsimps [sublist_upt_eq_take]
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addsimps [bag_of_sublist]) 1);
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by (rtac Follows_trans 1);
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by (rtac Follows_setsum 2);
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by (asm_full_simp_tac (simpset() addsimps []) 3);
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by (Clarify_tac 2);
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by (asm_full_simp_tac (simpset() addsimps [o_assoc RS sym]) 2);
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by (rtac (impOfSubs (mono_bag_of RS mono_Follows_o)) 2);
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(*Now need to invoke the merge_follows guarantee*)
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by (asm_full_simp_tac (simpset() addsimps [merge_spec_def,merge_follows_def]) 2);
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by (res_inst_tac [("f", "%s. bag_of (sublist (merge.Out s) {..length(merge.iOut s)(})")]
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Always_Follows 1);
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setsum (%i. if i < length l then {# nth l i #} else {#}) {..length (iOut s)(}
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: (%s. bag_of
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(sublist (merge.Out s) {..length (iOut s)(})) Fols
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(%s. setsum (%i. (bag_of o sub i o merge.In) s)
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{..Nclients(})
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(%s. sublist (merge.Out s)
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{k. k < size(merge.iOut s) & nth(merge.iOut s)k = i})
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Fols (sub i o merge.In))"
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by Auto_tac;
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by (dtac guaranteesD 1);
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by Auto_tac;
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by (rtac guaranteesI 1);
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merge_eqOut
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Always_Int_eq
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by (simp_tac (simpset() addsimps [o_def]) 1);
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by (subgoal_tac
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"ALL s. merge.Out s = sublist (merge.Out s) {..length(merge.iOut s)(}" 1);
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by (etac ssubst 1);
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by (asm_simp_tac (simpset() addsimps [bag_of_sublist]) 1);
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by (stac bag_of_sublist 1);
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